1.

Introduction

Power System

Power systems are large and complex

electrical networks designed to generate, transmit and distribute electrical

energy to different types of constantly varying loads. The transmission and

distribution of power is done through the alternating current (ac) system.

Since power loss is dictated by the formula I2R, transmission is

done at high voltage (HV) to increase transmission efficiency.

A well-designed power system has the

following characteristics:

1.

It can supply power practically

everywhere the customer demands.

2.

It can supply power to the

customer all the time.

3.

It can always supply the

ever-changing load demand.

4.

The power supplied is of good

quality.

5.

The power supplied is

economical.

6.

It satisfies the necessary

safety requirements.

The structure of the Mauritian power system

is shown in Figure 1.

Figure 1.1: Power System Structure in Mauritius (Source: CEB)

Power System Stability

Power system stability can be defined as

the ability of an electric power system to regain equilibrium in its state of

operation such that practically the entire system remains intact and all its system

variables stay bounded after being subjected to a physical disturbance.

Figure

1.2: Classification of Power System Stability

Voltage Stability

Voltage stability is the ability of a power

system to maintain steady voltages at all its buses after it is subjected to a

disturbance from an initial point of operation. Voltage stability is dependent

on the system’s ability to restore equilibrium between load demand and load

supply.

Some possible consequences of voltage

instability are:

1.

Loss of load in some parts of

the system.

2.

Tripping of transmission lines

leading to cascading outages.

3.

Loss of synchronism of

generators which may arise from the outages.

4.

Electric motors tend to run on

over speed when they are fed with higher voltages resulting in vibration and

mechanical damage. Over voltage may also cause insulation failure.

5.

Heating issues as a result of

drop in voltage and subsequent rise in current.

To avoid these nefarious effects, it is

important to keep the system voltage fluctuations to a minimum.

Voltage stability can be broken down into

two subcategories:

1.

Large disturbance voltage

stability

It refers to the

system’s ability to maintain steady voltages following large disturbances such

as system faults, loss of generation, or circuit contingencies.

2.

Small disturbance voltage

stability

It refers to the

system’s ability to maintain steady voltages when subjected to small agitations

such as incremental changes in system load.

Voltage Collapse

Voltage collapse is the process by which

the series of events associated with voltage instability leads to a blackout or

abnormally low voltages in a significant part of the power system.

Voltage Stability Timeframe

The timeframe of interest for voltage

stability can be divided into two parts, namely:

1.

Short-term voltage stability

2.

Long-term voltage stability

Short-term voltage stability

This involves the dynamics of fast acting

load components such as induction motors, electronically controlled loads, and

High Voltage Direct Current (HVDC) converters. The study period of interest is

in the order of several seconds, and analysis requires solution of appropriate

system differential equations.

Long-term voltage stability

This involves slower acting equipment such

as tap-changing transformers, thermostatically controlled loads, and generator

current limiters. The study period of interest is in the order of several

minutes. Long term simulations are required for analysis of system dynamic

performance.

Frequency Stability

Frequency stability refers to the ability

of a power system to maintain a steady frequency after a severe system upset

resulting in a significant imbalance between generation and load.

The resulting instability occurs in the

form of sustained frequency swings leading to tripping of generating units

and/or loads. In large interconnected power networks, the systems are split

into islands. Stability in this case depends on whether each island will reach

a state on operating equilibrium with minimal unintentional loss of load.

Generally, frequency stability problems are

associated with inadequacies in equipment responses, poor coordination of

control and protection equipment, or inadequate generation reserve. Frequency

stability may be a short-term phenomenon or a long-term phenomenon.

Frequency fluctuations can have the

following effects:

1.

Three phase ac motors run at

speeds that are directly proportional to the frequency. The variation of system

frequency affects the motor performance.

2.

The blades of stem and water

turbines are designed to turn at a pre-determined speed. Frequency variations

causes changes in that speed which results in excessive vibration and hence can

cause damage to the turbine blades.

3.

Frequency error may result in a

disaster in digital storage and retrieval process.

Automatic Voltage Regulators

Figure 1.3: Simple AVR System

Representation (Source: Saadat)

The Automatic Voltage Regulator (AVR) is

used to control the reactive power on the generation side and hence the

terminal voltage on the load side. The main duty of the AVR is to maintain the

terminal voltage of the synchronous generator at a predetermined level. Figure 1.3 shows

a simplified representation of an AVR system.

When the reactive power load of the

generator increases, the terminal voltage level suffers a drop in magnitude.

This variation in voltage level is sensed through the potential transformer on

one phase, rectified and fed to a comparator. The comparator compares this

feedback signal to a DC setpoint Vref and outputs an error signal to

the amplifier. The amplifier acts based on the error signal and increases the

terminal voltage across the exciter. This results in an increase in the

generator field current and hence an increase in the emf generated. The outcome

is an increase in the reactive power which in turn increases back the terminal

voltage to the desired level.

(Saadat,

2004)

Proportional Integral Derivative (PID)

Controllers

A PID (Proportional Integral Derivative)

controller is a popular, simple-to-use control algorithm used in the industry

to overcome various kinds of problems. Figure 1.4 shows

the PID controller in block form.

Figure 1.4: PID Controller

Representation

The PID controller can be represented

mathematically by the following equation in the s-domain (Laplace operator):

Or

Where

Kp is the proportional

constant term

Ki is the integral constant

term

Kd is the derivative constant

term

Ti is the integral action time

(also known as reset time)

Td is the derivative action

time (otherwise known as rate time)

A more general equation for the output of

the PID controller in time domain is:

Where u(t) is the controller output signal

and e(t) is the error signal.

Ant Colony Optimisation Algorithm

The Ant Colony Optimisation (ACO) algorithm

is a swarm intelligence based metaheuristic technique which mimics the

behaviour of ant species looking for food. A trail of pheromone built up by

previous ants on their way back from the food source to the colony helps the

ensuing ants to locate the best route to their food destination. Every ant

contributes to find a solution to the given problem; whereas the whole colony

work together to find the optimal solution. ACO has been used in many difficult

combinatorial optimisation problems such as the travelling salesman problem

(TSP), quadratic assignment problem, hydroelectric generation scheduling

problems, inter-alia. (Dorigo, et al.,

2006)

Whale Optimisation Algorithm

The Whale Optimisation Algorithm (WOA) is a

metaheuristic optimization algorithm that draws inspiration from the

predatorial movements of humpback whales in locating and hunting down their

preys. This behaviour, demonstrated in Figure 1.5, is

specifically observed in humpback whales and is termed as the bubble-net

feeding method in which the whales create bubbles along a spiral-shaped path

surrounding their prey. A mathematical model following the bubble-net feeding

method is derived in WOA to perform optimisation. (Mirjalili, et al., 2016)

Figure 1.5: Bubble-net feeding

behaviour of humpback whales (Mirjalili, 2016)